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This is a classic topic that seems simple at first glance but is actually a lot more complicated: What keeps trains on tracks? Everyone has seen train tracks and train wheels so you know what they look like. The simplest and most obvious answer is that the flanges (the wider parts on the insides of the wheels) keep the trains on tracks. Partly yes, but the flanges are actually just a safety feature that keeps the train from slipping sideways off the track by accident. When you consider a train going 160 km/h (99.4 mph) you can imagine the screeching that the flanges would create if they were in contact sideways with the tracks. So normally the flanges don’t touch the tracks at all.

What really keeps a train on tracks is actually pretty clever: The wheels are not cylindrical but conical. The tops of the rails are a bit convex and the rails are also not strictly vertical, but they are canted the same amount inwards as the wheels are conical. The convex rail causes the wheels to touch the rails only at a single point. The image below hopefully helps to illustrate the geometry:

Train wheels and tracks when turning

Because the wheels are fixed to a rigid axle, they rotate at the same pace. When the tracks start to turn (left in this example), the train tries to go straight and the outer (right) wheel’s point of contact with the rail moves inwards. The inner wheel’s point of contact in turn moves outwards. Now because the wheels are conical, the wheels’ effective circumference (circumference at the point where the rail touches the wheel) is no longer the same, but instead the outer wheel travels longer distance each rotation than the inner. This makes the outer side move forward faster than the inside, which turns the train back to where the tracks lead.

Real tracks are also not level with each other in curves, but instead the outer rail is higher than the inner. This helps the train to turn and allows for higher speeds. Train tracks usually have long turn radii too, so the trains don’t need to turn that fast. How about trams then?

Trams usually share the street space with cars, bikers and pedestrians so canted trackbed is usually out of the question. In a city you can’t have turning radii of hundreds of meters either so trams need to be able to make sharper turns. This is why trams cheat a bit.

A train’s wheel will most likely break if the train runs on the flange, but tram wheels are actually built so that they can do that. This helps in sharp turns, when the outer wheel can transition to run on the flange instead of the regular rim, which makes that wheel’s effective circumference a lot longer than the inner wheel’s, thus turning the tram faster.

All of this is of course a simplified version about what’s happening. There’s a lengthy explanation in Wikipedia about Hunting Oscillation (that is still a simplified version instead of a full analysis) which explains what kind of forces act on train wheels and tracks and how they interact. The page is really worth a look since it has a really nice animated image that shows how a wheel set moves around. I have to warn you though that the train tracks rabbit hole in Wikipedia is deeeeeeeep.

The recent years have brought us lots of news about new exoplanets. A lot of the discoveries are credited to NASA’s hard working space telescope Kepler. The way Kepler works means that the first new planet candidates were primarily orbiting really close to their parent stars. You have probably seen lines like “the planet XYZ is orbiting so close to it’s host star ZYX that it is tidally locked and always turns the same side towards the star” in the news about the discovered stars. What’s this tidal locking thing then? How come a planet that orbits freely around a star has its rotation synchronized to the planet’s motion around the sun?

Turns out we don’t need to look at other solar systems to find other examples of tidal locking. The rotation of our very own moon is tidally locked, showing the same side towards us all the time. Mercury, the innermost planet in our solar system is also tidally locked to sun, but it exhibits a more complex 3:2 ratio (so it rotates three times around itself for every two orbits around the sun) because of the eccentricity of Mercury’s orbit.

But back to the issue here: how does all of this happen and what’s the connection with tidal locking and tides (they are indeed related)? To understand what happens, we need a little bit of physics:

Let’s go back in history to a time when the moon was still rotating around its axis so that an observer on earth, if there would have been one, would have seen all the sides of moon would he have stared long enough. To keep things simple, let’s assume that the moon orbits the earth on a circular orbit and that the only force in effect is earth’s gravity pulling the moon towards earth. We know that the gravitational force is proportional to the product of the masses interacting and inversely proportional to the square of distance between them.

The second part is actually important here: if we think of the moon as two halves of a sphere, cut along a plane perpendicular to the earth’s gravitational pull, we can see that earth actually pulls the different halves with a little bit different force. The farther half of the moon is – well – farther away and thus feels a slightly smaller force. This unevenness causes the moon to deform from an ideal sphere and get elongated along an axis pointing towards earth.

Now that the moon is spinning, the bulge is actually not pointing straight towards earth. The moon rotates all the time and since it’s actually made of rock, it resists the deformation. This makes the bulge shift a bit to the side where the moon is rotating.

Now that the moon is shaped like an american football and it does not point straight to earth, the interesting part happens: The bulge closer to the earth feels a bigger gravitational pull than the bulge on the other side of the moon. Since the forces are not aligned along the axis that points towards earth, the net torque isn’t zero. This makes the gravitational pull of earth actually slow down the rotation of the moon until the bulge is aligned with the gravitational pull, ie. the orbiting body is tidally locked.

This actually happens to the earth too: same mechanism causes earth to deform and tides in oceans. The bulges created by moon’s gravity are not exactly in sync with the moon either, so the moon slows down our rotation. This loss of angular momentum is transferred to the moon which lifts its orbit and makes it get farther away from us. At current pace, the moon’s orbital radius grows by 38mm per year and our day gets 15 microseconds longer each year because of the slowing down.

All of this because of a force between two bodies in the emptiness of space.

When I was thinking about what the header picture for this blog should be, I almost immediately thought of the fractal cabbage. The vegetable is really named Romanesco Broccoli, and the complexity of it just blows your mind. It’s related to broccoli and cauliflower and is of a vibrant, almost neon green color. The most amazing thing about Romanesco Broccoli is that it resembles a natural fractal. The bud is made of a logarithmic spiral of cones, each one itself a spiral of cones, each one… Well, you get the point. It seems that the self-similarity goes on down to a level where you can actually use a scanning electron microscope. There are a few excellent pictures of the fractal cabbage in different scales in this article.

It also turns out that if you start counting the spirals in the bud in clockwise and counterclockwise direction, those numbers are always neighbouring Fibonacci numbers. A great (and lengthy) explanation why this happens (actually quite a lot in the nature) can be found here.

So, the most amazing vegetable ever! It’s a fractal! It looks like something that was computer generated! And there are Fibonacci numbers involved!

So, a new blog project, again. I’ve wanted to write something for a long time and finally actually found something that seemed plausibly interesting so that I might see my short attention span keeping up with writing for a longer time. To see what this whole thing is all about, visit the about page.